Suppose we have a set of regular functions defined on a product of metric spaces, for instance the Banach space of the smooth functions from $\mathbb R^n$ to $\mathbb C$. We know, thanks to the Taylor series, that such a function can 'almost' be written as a linear combination of products $f_1 \cdot \ldots \cdot f_n$, each of the $f_i$ a continuous function from $\mathbb R$ to $\mathbb C$, where the almost means we are using a density argument in the sup norm.

Is there a way to generalize such a situation to a generic product of metric spaces? In particular I am interested in the space of locally constant, compactly supported functions on powers of the $p$-adic field $\mathbb Q_p$ but also the general question sounds interesting. Any reference is greatly appreciated, thanks.

The Taylor series of a function doesn't always converge to it in the sup norm even in one dimension (take $e^x$). What exactly is the precise statement you want?
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Qiaochu YuanJul 2 '12 at 15:28

A reasonable construction (for Banach space-valued functions defined on a product of metric spaces) is via partitions of unity. Given a continuous function $f:X\times Y\rightarrow (E, \|\cdot\|)$, and $\epsilon >0$, there is a uniform approximation $\epsilon$-close to $f$ of the form $f_\epsilon(x,y):=\sum_{i,j} f(x_i,y_j)\phi_i(x)\psi_j(y)$.
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Pietro MajerJul 2 '12 at 17:59

1 Answer
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Is this the statement you want: any locally constant, compactly supported function from ${\bf Q}_p^n$ to ${\bf C}$ is uniformly approximated by linear combinations of products $f_1\cdots f_n$ where each $f_i$ is a locally constant, compactly supported function from ${\bf Q}_p$ to ${\bf C}$? Yes, this follows from the Stone-Weierstrass theorem for locally compact Hausdorff spaces. The linear combinations in question constitute a self-adjoint algebra of functions which separate points and separate each point from infinity. Therefore they are uniformly dense in the space of continuous functions vanishing at infinity on ${\bf Q}_p^n$.